Thursday, October 12, 2006

Mathematics & Game Design, Part One

I've written more than once that I don't like game designs that require me to do math. More specifically I'm talking about games like Santiago, Power Grid, and several others which have a strong mathematical basis and a strong ability to analyze that math during game play.

You see, I think games should be entertaining: it's why I play them: to enjoy myself and to have fun. And, I don't have a lot of fun when I sitting around adding, multiplying, and dividing (except, perhaps, in the case of a game of Primordial Soup).

Worse, I increasingly think that games which have a strong mathematical component have a core flaw in them related to victory. This flaw comes about because there are generally three types of players who might be playing these games:

Some players will be totally oblivious to the underlying mathematical basis of the game, and will play by gut because it's their only way to do things.

Some players will understand the mathematical basis of the game, but will choose to largely or entirely ignore it because it detracts from their fun to carefully figure things out.

And finally, some players will embrace the math, carefully calculating and recalculating every move against the mathematical basis which is laid bare.

Now the flaw arises from the fact that player type #3 will generally win these games if the underlying mathematics are actually sound. Worse, he'll probably do so by minutely analyzing all of the options, each turn, to the point where his turns might take two or three times as long as his opponent's. I don't find it particularly endearing for a game's main path to victory be raw time put into the game, yet for many mathematically based games, that's exactly what happens.

Now don't get me wrong, I don't think that the flaw is having mathematics at the basis of a game. On the contrary, that's often good game design as mathematician Reiner Knizia has proven through many a game. Instead the problem is placing the mathematics so close to the surface, and making them so static--so unchangeable--that you can calculate them without any fudge factor, revealing exactly the valuation of any move.

Auction games are in particular danger of hitting this pitfall, since you're expending limited resources in order to gain victory in some form. Things get even more dangerous when you're actually making your purchases with the commodity that will ultimately be used for victory points, as this allows for a simple apples-to-apples comparison if the math lies too near the surface.

However, there are ways to combat all of these problems. One of the easiest answers is to introduce chaotic player interactions, so that valuations depend upon the actions of other players. Another is to give players more explicit ways of changing valuations. A third is to introduce a few levels of abstraction between purchase and victory.

A few different games reveal how these methods can work, and when they didn't ...

Santiago: In this game a set of plantation tiles is revealed, and then players bid for selection order among that lot with money (victory points).

The problem for me arises in the very mathematical results of the tile placement. Whenever he makes a placement a player scores an amount equal to the number of his control markers times the number of connected plantation markers of the same type.

Thus it's pretty easy to make a calculation that goes like this: "There's just one good tile for me, and that's the double-marker banana. If I purchase it the overall banana field goes from size 6 to 7, and my number of control markers goes from 2 to 4. Thus the tile represents a 4x7-2x6=16 point gain for me. The next best tile (after carefully running through calculations for all of them) just nets me 6 points, so going first represents a 10 point gain. Thus, if I bid 10 I breakeven. The bid's gotten to me, and it looks like I need to bid 4 to take it. I guess that's worthwhile, since it's the same game as my second choice."

Now granted, there's chaos here, because other players going after you could mess up your calculations, and a tile can become worthless if water doesn't flow to it (though as a first player, you'll also often have the ability to place where there's already water). Nonetheless, the math is so close to the surface, that it seems pretty clear that a careful calculator will do better than a gut-feeler.

Don't get me wrong, I think Santiago is a fine game, with quite a few interesting gameplay elements. However, I can only play Santiago when I'm pretty tired, because when my brain is fuzzy I don't automatically start doing all the calculations. When I'm tired enough, Santiago is fun.

Boomtown: In this game, a set of cards are revealed, and then players bid for first-place selection among them with money (victory points). Last-place selectors also get rewarded with some bonus cash.

On the base level, this game's a lot like Santiago. You're bidding with victory points (money) for victory points that are pretty directly related. However after that it goes further afield in some good ways.

First, Boomtown doesn't require the same type of calculation as Santiago. You're bidding on mines, and the value of each mine is printed on the mine. However, each mine also has a secondary value: it can produce money (victory points) during the game if it "produces", and that's determined by a 2-12 roll each turn.

Now, I suppose you could try and calculate a mine's valuation if you wanted:

However, probability of this sort just isn't as natural of a calculation as the simple multiplication of Santiago. I don't have to stop myself from doing it, and I suspect it doesn't even cross the minds of most folks. (It really didn't cross my mind until I started in on this article.)

Better, Boomtown further obfuscates value by giving additional points for majority control of the five different colors of mines. This results in both short-term payments from other players and a payout from the bank at the end. Thus, certain colors of mines are much more valuable to certain players, but the exact amount of that value isn't clear.

This sort of inclarity (or abstraction, to paint it in a more positive light) can really benefit a game because it takes away the advantage of the math-hounds. Sometimes Boomtown is still a little too mathy for me, but I'm more likely to play it and enjoy it when I'm not tired.

Ra: In this games, sets of tiles are revealed, and then players bid for entire lots with special "sun" tiles.

First, it's clear that Ra does have a mathematical basis. You just have to look at the list of scores for different items, which vary widely, and were probably very carefully considered, to see that. However, Ra does a lot to abstract those tile valuations by introducing multiple levels of uncertainty.

First, unlike both Santiago and Boomtown you're not bidding with victory points. Instead, your sun tiles affect your buying power in future rounds, but in a way that's not entirely direct. At the end of the game sun tiles are turned in for victory points, but in pure competition to other players, a competition where you won't know your standing until the last moment unless you really carefully monitor all the plays.

Second, there are a few tiles that have direct and immediate point values (gold and a first civilization), but other tiles have solely speculative value (additional civilizations, monuments). You're counting on being able to make additional purchases later in the game in order to give them value, and again while this valuation could be determined to some extent by a probabilistic calculation, it's not ever done.

Third, some victory points (for pharaohs and final money) come about through player competition, and as with the colored-mine competition in Boomtown, this introduces a lot of uncertainty into the calculation.

Fourth, I think Ra makes a very good move by giving people the ability to buy an entire set of tiles rather than singletons. With a singleton purchase the human instict is much more to try and make a valuation, but with a lot everything starts to get lost in the static, and you just start looking at the high points and low points.

And then you make a decision by gut.

Some people complain about the randomness of Ra, and this goes to the exact strengths that I see. There is uncertainty, granted--chaos and speculation--but it's that same thing that keeps the game from becoming a number-crunching exercise that you could set up on a spreadsheet, where he who takes the longest turn wins.

Next Time: Though I think math should largely be hidden from players, designers absolutely need to think about it. In my next column on this topic I'm going to talk about a few game designs that failed the sound-underlying-mathematics test, and I'm even going to show the math why.

Interesting article. I know that I actually enjoy doing the math in many games, and that's why I like them. But then, I was a Math and Statistics major, so obviously doing calculations is very enjoyable for me.

That's definitely a lot of why I like Power Grid. I like doing all the calculations every turn -- "I can bid this much right now, because I have to save that much to buy 4 coal and 3 oil, and I'm going to need to build 3 routes this turn." I think I can do the calculations fast enough, while other people are bidding, that I don't slow the game down. I certainly try to play at a good speed.

Conversely, this is why I'll veto games like Torres and Java. The game screams out to be calculated, but it's just not possible. There are too many darn choices to make, but all of them could be computed to figure out which one is the best one. And it just turns out to be frustrating to me.

And it's funny, I think the one time I played Boomtown (with you), I was basically doing the calculation of value that you describe.

Maybe you just don't like the byproduct of a game that makes you do math? For many games the math involved is absolutely necessary for the game design. The game would not be a good game at all if it did not have a foundation in mathematics.

Usually the byproduct of a game that "makes you do math" is the analysis paralysis that inevitably happens especially near the end of the game.

A few good solutions to cure this problem. Hidden resources and money. Powergrid for example is typically played with hidden money to cut down on the AP time. Also using a timer is also a good way to limit a player taking forever to make their decision.

I have a decent mathematical background but I am perfectly willing to "spitball" (estimate) my decisions in order to speed the game along. I admit I do get caught in the AP trap and will start calculating things especially when others shamelessly do it.

If game designers were not allowed to design games with solid mathematical foundations, I think that would make for a lot of crappy games being produced. Or maybe they would all be RPG's which would probably make some people happy. But even RPGs have a mathematical basis.

If you look you'll never find me
I've gone gaming I'll be back
Got to ship a few more barrels
Got to trade a brick or two
Got to buy some more provisions
Got to run this train on through
Got to roll a few more 6's
Got to draw a few more cards
Got to shake hands with my neighbours
'Fore I'm back in my backyard
But if you look up to the night sky
That's my spaceship passing by
Give a wave that's me inside her
It's farewell but not goodbye